# Adaptive multilevel stochastic collocation methods for uncertainty quantification

Lead Research Organisation:
University of Birmingham

Department Name: School of Mathematics

### Abstract

Computer simulations in science and engineering rely on mathematical models of the underlying phenomena and processes. These mathematical models are typically written in terms of partial differential equations (PDEs) relating rates of changes of physical quantities (e.g., temperature in a solid or velocity of a flowing fluid) in space and time. Realistic models of complex phenomena and processes must account for the ever-present uncertainties resulting e.g. from imprecise or incomplete knowledge of all inputs to a PDE-based model (such as material properties, initial conditions, external forces, etc.). Examples of such phenomena include wave propagation in inhomogeneous media with uncertain wave characteristics and fluid flow through a porous media with permeability not known precisely at every point in the computational domain. In these cases, instead of standard deterministic models, simulations must rely on probabilistic techniques in order to model the underlying uncertainties in the inputs (using random variables or random fields), analyse how the uncertainties propagate to the model outputs, estimate probabilities of undesirable events (e.g., the contamination of groundwater resulting from a leakage from nuclear waste repository), and perform reliable risk assessments. The models are then represented by PDEs with random data, where both inputs and outputs take the form of random fields.

The development of effective approximation techniques and numerical algorithms for solving PDEs with random inputs is an important task in uncertainty quantification, because it opens the door to realistic simulations and ensures reliable and accurate predictions in the presence of uncertainties. Key mathematical challenges in this research area concern (i) the design of approximation methods with guaranteed and reliable error control, and (ii) the development of provably accurate adaptive algorithms that make the best use of available computational resources. This project will address both aforementioned challenges by developing, analysing, implementing and testing a novel methodology for reliable error estimation and adaptive error control in the framework of a powerful approximation technique for PDEs with random inputs known as the multilevel stochastic collocation finite element method. The project is relevant to many applications in engineering and manufacturing (e.g., in nuclear power industry) where improvements in the efficiency and reliability of numerical methods for uncertainty quantification would speed up decision making and have a direct impact on public safety.

The development of effective approximation techniques and numerical algorithms for solving PDEs with random inputs is an important task in uncertainty quantification, because it opens the door to realistic simulations and ensures reliable and accurate predictions in the presence of uncertainties. Key mathematical challenges in this research area concern (i) the design of approximation methods with guaranteed and reliable error control, and (ii) the development of provably accurate adaptive algorithms that make the best use of available computational resources. This project will address both aforementioned challenges by developing, analysing, implementing and testing a novel methodology for reliable error estimation and adaptive error control in the framework of a powerful approximation technique for PDEs with random inputs known as the multilevel stochastic collocation finite element method. The project is relevant to many applications in engineering and manufacturing (e.g., in nuclear power industry) where improvements in the efficiency and reliability of numerical methods for uncertainty quantification would speed up decision making and have a direct impact on public safety.

### Organisations

## People |
## ORCID iD |

Alexey Bespalov (Principal Investigator) |

### Publications

Bespalov A
(2023)

*Error Estimation and Adaptivity for Stochastic Collocation Finite Elements Part II: Multilevel Approximation*in SIAM Journal on Scientific Computing
Bespalov A
(2022)

*Error Estimation and Adaptivity for Stochastic Collocation Finite Elements Part I: Single-Level Approximation*in SIAM Journal on Scientific ComputingDescription | Partial differential equations (PDEs) with parametric or uncertain inputs are ubiquitous in the mathematical modelling of many phenomena and processes in science and engineering. They naturally occur in simulations of systems depending on many parameters and in practical applications with inherent uncertainties where a complete and precise characterisation of all inputs (e.g., material properties, initial conditions, external forces) is not possible. In these cases, simulations must rely on probabilistic techniques in order to model the underlying uncertainties in the inputs, analyse how the uncertainties propagate to the model outputs, estimate probabilities of undesirable events, and perform a reliable risk assessment. The models are then described by PDEs with random data, where both inputs and outputs take the form of random fields. Numerical solution of such PDE-based models is significantly more challenging than the solution of the deterministic analogues. The project was focused on a particular numerical method for PDEs with parametric or uncertain inputs -- the sparse grid stochastic collocation finite element method (SC-FEM). The main objectives of the project concerned the development and implementation of new adaptive multilevel stochastic collocation algorithms that are underpinned by a rigorous a posteriori analysis of underlying approximation errors. These objectives have been achieved, resulting in 2 journal publications and 1 software package. In particular, the work on the project followed three inter-related strands. (1) A posteriori error analysis of approximations. Rigorous a posteriori error analysis of computed solutions is at the heart of adaptive algorithms for PDE problems. We have developed novel reliable a posteriori error estimators and practical error indicators for single-level and multilevel SC-FEM approximations for general diffusion problems with uncertain permeability. (2) The design of adaptive algorithms. We have exploited the developed a posteriori error estimators and indicators and designed a fully automated procedure that ensures proper interplay of spatial and stochastic components of multilevel SC-FEM approximations. The algorithms start with a small number of collocation points (typically, a single point) and a coarse finite element mesh. Guided by theoretically justified error indicators, the algorithms incrementally enrich the set of collocation points and refine the underlying spatial meshes individually for each collocation points. The total error in the generated SC-FEM approximations is thus controlled and reduced until the prescribed tolerance is reached. (3) The implementation of algorithms and software development. Software development and numerical experimentation have been an integral part of research on the project. We have implemented the developed algorithms in an open-source MATLAB toolbox Adaptive ML-SCFEM that is freely available to other researchers. |

Exploitation Route | There are many possible extensions to the results of the project in terms of analysis, algorithmic developments and implementation. These include goal-oriented error estimation and adaptivity, convergence analysis of the developed adaptive algorithms, extensions to more complex PDE problems stemming from the uncertainty quantification models of practical interest. The open-source software package created within the project provides researchers with computational tools for experimentation and exploration, enabling them to apply our algorithms to other problems and test alternative algorithms. |

Sectors | Education Energy Environment |

URL | http://web.mat.bham.ac.uk/A.Bespalov/papers/index.html |

Title | Adaptive ML-SCFEM |

Description | Adaptive ML-SCFEM is a MATLAB toolbox for computing adaptive stochastic collocation finite element approximations for elliptic PDEs with random inputs. This is free software. It can be redistributed and/or modified under the terms of the GNU Lesser General Public License. |

Type Of Technology | Software |

Year Produced | 2023 |

Open Source License? | Yes |

Impact | Adaptive ML-SCFEM is a MATLAB toolbox for computing and investigating adaptive stochastic collocation finite element approximations for diffusion problems with random coefficients. The software has been used to generate numerical results in the papers on adaptive stochastic collocation FEM. It provides researchers with computational tools for experimentation and exploration, enabling them to apply the algorithms to other problems and test alternative algorithms. |

URL | https://github.com/albespalov/Adaptive_ML-SCFEM |